Physica 19D (1986) 89-111
North-Holland, Amsterdam
STEADY SOLUTIONS OF THE KURAMOTO-S1VASHINSKY EQUATION
Daniel MICHELSON*
The Hebrew University of Jerusalem, Institute of Mathematics and Computer Science, Givat Ram, 91904 Jerusalem, Israel and
University of California at Los Angeles, Department of Mathematics, CA 90024, USA
Steady solutions of the Kuramoto-Sivashinsky equation are studied. These solutions are defined on the whole x line and
propagate with a constant speed c z in time. For large c 2 it is shown that the solution is unique and has a conical form. For
small c 2 there is a periodic solution and an infinite set of quasi-periodic solutions as asserted by Moser's twist map theorem.
Numerical computations for intermediate values of c 2 suggest that below c 2 = 1.6 for every speed there is a continuum of odd
quasi-periodic solutions or a Cantor set of chaotic solutions wrapped by infinite sequences of conic solutions.
Received 14 September 1984
Revised manuscript received 8 July 1985
v(x, t) although irregular has an appearance of a
1. I n t r o d u c t i o n
The Kuramoto-Sivashinsky equation
u,+ vau+ v2u+XlxTul2=O, u=u(x,t)
(1.1)
has attracted for the last decade a considerable
attention [1-12]. It was originally derived by
Kuramoto and Tsuzuld [1] in the context of a
reaction diffusion system, and by Sivashinsky [5]
in the context of flame front propagation, In the
later case u(x, t) represents the perturbation of a
plane flame front which propagates in a fueloxygen mixture. Numerical experiments [2, 3, 6]
have shown that eq. (1.1) when solved on a sufficiently large interval - l < x < l with periodic
boundary conditions tends to a turbulent state as
t --, oo. The solution u(x, t) has the form
u ( x , t ) = - 4 t + o(x, t),
(1.2)
where c02 ~ 1.2 is a universal constant independent
of the initial condition, while the mean value of
v(x, t) is close to zero. For a fixed t the function
*This work was supported by an Alfred Sloan Foundation
Fellowship and the National Science Foundation under Grant
No. MCS 78-01252.
quasi-periodic wave with a characteristic wave
length 1o = 2~r/%, w0 = v~-/2. Note that the
frequency % is maximally amplified by the linear
terms in (1.1). Formula (1.2) suggests that one
should look for steady solutions of (1.1)
u(x, t) = -c2t + o(x).
(1.3)
Clearly o(x) satisfies the O.D.E.
da---~u+ --dZv= c 2 - l[d°~2
dx 4 dx 2
2 ~ dx J
(1.4)
or a third order equation for the derivative y =
do/dx
d3y
d y = c2
1
2
-oo<x<+oo.
(1.5)
The main objective of this work is to study the set
of bounded solutions of (1.5) and its dependence
on the parameter c.
For large c we shall show that eq. (1.5) has a
unique (up to translation) bounded solution. This
solution y(x) is an odd function of x, tends to the
limits limx_. ±~ y(x) = q:c~- and vanishes only
at x = 0. The integral v(x) = f0~y(r)dr has thus a
0167-2789/86/$03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
90
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
conical form with a single maximum at x = 0 and
slopes + v ~ c as x ~ q: oo. The above function
o ( x ) has a following physical interpretation. A
slight modification of (1.1),
u,+ ~74u+ ~72u+ ½1~7ul2= c 2,
i.5--
P3
P2
LO"
(1.6)
is a model equation (due to Sivashinsky) for a
conical flame front with a slope c ~ - on a Bunsen
burner. Clearly, the above o ( x ) represents a stationary flame on a Bunsen burner.
F o r small c eq. (1.5) has a periodic solution Yp~r
with frequency to depending on c. It turns out that
the Poinear6 map associated with the periodic
solution is measure preserving and satisfies for
small c the conditions of Moser's twist map theorem [14]. As a result, the flow defined by (1.5)
possesses an infinite set of coelecial invariant tori
surrounding the periodic orbit. The boundaries of
the tori are closures of quasi-periodic orbits. We
show that the integrals of these quasi-periodic
solutions are quasi-periodic too. This results in an
infinite set of quasi-periodic solutions of (1.4).
The periodic solution Yper as well as to and c
could be expanded in power series with respect to
an auxiliary parameter e. The numerical computation of these series reveals that the periodic
solution exists up to Cm~x -- 1.26606. The complete
to, c curve is displayed on fig. 1. The part PoP2Q of
the graph corresponds to the domain where the
analytic expansion is valid. The portion QPhP8 was
calculated using a difference approximation to
(1.5). The branch P4P[Pff represents a non-odd
periodic solution bifurcating from the point P4.
This branch could be continued until the limit
point w = 0, c---0.86 which corresponds to an
"oblique" soliton (see figs. 4c-4e). The periodic
orbits are parabolic at the points Pi, P/, and elliptic
and hyperbolic at the segments indicated by letters
e and h respectively. In the elliptic regions the
Moser's twist map theorem applies so that there
are infinitely many invariant tori.
A central difference approximation of (1.5) was
used in order to compute the bounded odd solutions for intermediate values of c. The results
suggest that for c > c 1 -- 1.3 there is only one such
P~
0.5"
e
6
Yk
a
]PP8
0.0
I
0,3
','~
0,4
....
I'B"'~
0.5
0.8
....
I ....
0.7
I ....
O.B
J'"
0,9
i
P0
aDll]
i,O
I.!
OM
Fig. 1. The to-c curve for the periodic solution. The branch
P4P~P~ corresponds to the non-odd periodic solution.
solution. In the interval Cm~x < c _< c 1, there is a
decreasing sequence of bifurcation points
c 1,c a, . . . , c n-~cma ~. At c = c ,
a new solution
y , ( x ) is " b o r n " which splits into two solutions as
c decreases. The function y , ( x ) tends to the critical points +_cv~ as x--* -T-oo and has n zeros in
the half line x > 0. Thus the integral o , ( x ) =
fo~y,('r)d'r would correspond to a Bunsen flame
with n + 1 maxima. At c = Cmax the above solutions tend to the periodic one as n ~ oo. Fol
c < c,,~x the set of odd bounded solutions (of th~
scheme) is quite complicated. The basic form o!
the solutions is determined by the periodic orbits
In the elliptic domains there are invariant tor
while in the hyperbolic domains there exists ;
Cantor-type set of chaotic solutions which "float'
around the periodic ones. On the other hand fo
all values of c below Cm~x (at least up to c = 0.2
there exist infinite sequences of odd asymptoti
solutions (i.e. connecting the critical points). Thes
sequences of "Bunsen flame" type solutions lie i:
the "holes" of the Cantor set and approximate th
above chaotic solutions.
2. Topological properties of the set of
bounded solutions
First we shall prove that for any closed interv~
c ~ [0, c,] the sets of all bounded solutions of (1.'
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
91
are uniformly bounded. Rewrite (1.5) as a first
order system
Proof. Assume to the contrary that there exists a
sequence of bounded solutions y~(x) such that
dP
t 2
d x = f ( Y ' c) = (Y2, Y3, c2 --1'2- ~Yl ),
SUPxlY~(x)l ~ oo. Introduce the norm IIYlI-E~_ t [y (o1 ~/~,. Without loss we may assume that
Y = ( Y l , Y2, Y3)" (2.1)
At that point it is worthwhile to consider a more
general situation of a system
dy
=/(Y)'
- oo < x <
Ily~(0)ll = P~ ~ oo
and
supJly~(xll/& <-2.
X
(2.8)
Change the variables z = O;'y, ~ = O;rx. Then the
function z~(~) satisfies the equations
(2.2)
dz n
d~ =¢P"(z")=h(z")+P~P~sg(P~z")'
where
y t N ) ) ~ R N, f =
(fo) ..... ftN))~ RlV and f ( y ) is a smooth vector
function, say f ~ CI(WV). Assume that
(2.9)
y = (ym .....
while
[Iz,(0)[I---1
and
supllz,(~)ll~2.
(2.10)
(2.3)
f(y)=h(y)+g(y),
where h(y) is a "pseudo-homogeneous" leading
part of f(y). Namely, there exists a positive real
vector s = (s 1, s2,..., s) and real scalar r such that
p'p-*h(p*y) = h(y)
(2.4)
and
Clearly, in the bounded set Ilzll< 2 the term
Oro-Sg(#Sz) tends uniformly to 0 as 0 ~ oo. Since
[Iz,(~)[I and Ildz,(~)/d~ll are uniformly bounded,
there exists a subsequence of zn(~ ) which converges uniformly on any finite interval to a
bounded solution z(~) of the equation d z / d ~ =
h(z). In view of (2.10) IIz(0)ll--1 which contradicts our assumption about the system (2.7).
prp--,g(p,y) ..., 0 as p ~ 0,
uniformly for
[y[ < 1.
(2.5)
Here
OSy = ( OS~yO), pSNy(N)),
prp-s h = (prp-hh(t) . . . . . Orp-SNhtN))
(2.6)
and similarly in (2.5).
Assume that the leading system
dy
dx =h(y)
Lemma 2.1 obviously applies to system (2.1)
with h ( ~ ) = (Y2, Y3, - -X2y2), s = ( 1 , 4 / 3 , 5 / 3 ) and
r=-1/3.
The system d y / d x =h(.p) is equivalent to the single equation y " = - ½y2 for y =
Yt- One can easily show that the last equation does
not have bounded solutions. Although the function f in (2.1) depends on a parameter c, it is clear
that in a compact domain of y and c variables,
the function g(fi) = (0, 0, c 2 -Y2) satisfies the condition of (2.5).
For c >> 1 change in (1.5) the variables
(2.7)
does not have bounded solution for - oo < x <
but y = 0.
Lemma 2.1. Under the above assumptions the set
of bounded solutions of (2.2) is uniformly bounded
in the maximum norm.
z=y/(cf2), ~=x(cv~) 1/3.
(2.11)
Eq. (1.5) then becomes
d3z
+
dz
=
1
(1 -
<< 1
(2.12)
D. Michelson/ Steady solutions of the Kuramoto- Sivashinsky equation
92
Theorem 2.1. Critical points or hyperbolic peri-
or as a system
d~
d--~ " ~ ( Z 2 ' Z 3 ' ~ ( 1 - - 7 " ? ) - - e T " 2 ) '
odic orbits may not be isolated components of the
set of bounded solutions of (2.14) for t > 0.
=
(2.13)
Again, as in system (2.1), the set of the bounded
solutions of (2.13) for e in a compact interval
e E [0, e,] is uniformly bounded.
Our next step is to show that the Conley index
of the set of all bounded solutions of (2.1) is zero.
Let us recall briefly (for details see [17]) some
properties of this index.
a) Conley's index is a homotopy type of a
pointed topological space.
b) For each isolated invariant set of a flow there
is a corresponding index.
c) The index of a disjoint union of isolated
invariant sets of a flow is a sum of their indices
(i.e., the homotopy type of the wedge of the corresponding pointed topological spaces).
d) The index of an isolated invariant set does
not change under a homotopy of the flow (provided the invariant set remains isolated under the
homotopy).
e) The index of a hyperbolic critical point or of
a hyperbolic periodic orbit is non-zero.
The flow in (2.1) could be extended by a two
parameter homotopy
dy
dx
= (
t-
to
d x = (Y2' Y3,
Theorem 2.2. There exists a constant e0 > 0 such
that for all [el < e0 system (2.13) has one and onl)
one (up to translation) bounded solution ~(x; e)
This solution connects the critical points ~L and
za, Zl(X; e) is an odd function of x and vanishe.,
only at x = 0.
Proof. The transversality implies that for small ,
-
[0,11 (2.14)
dy
Now we consider eq. (1.5) for large c or equivalently (2.13) for small e > 0. For e = 0 system
(2.13) was firstly studied in [18] (see also [19] and
[17]). Clearly it has a non-trivial bounded solution.
Since there is Liapunov function L ( ~ ) = z2z 3 z l / 2 + z~/6, the above solution connects the critical points. Recently McCord [20] has shown that
there is only one non-trivial bounded solution.
Hence the solution is odd, i.e. z = Zl(X ) is an odd
function. Moreover, Zl(X ) vanishes only at 0 and
dza/dx(O ) < 0. It is then easy to show that the two
dimensional stable manifold Mst(ZR) of the critical point ~R = (--1,0,0) and the unstable twodimensional manifold Mu(~L) of the critical point
~L = --~a intersect transversally along the above
bounded solution. We claim that the same results
hold also for small e 4: 0. Namely,
-c 2
- ~Yaz)-
The last system does not have bounded solutions
(see [17], p. 12). Denote by I(t, s) the set of all
bounded solutions of (2.2). Again as in (2.1) I(t, s)
is uniformly bounded (and thus isolated) for t and
s as in (2.14). Since the index of I ( - c 2, 0) = ~ is
zero, so is the index of l(t, s). Note that for t > 0
system (2.14) has two critical points YL = (2V~, 0, 0)
.and Ya = --YL, which are both hyperbolic. Thus
we have proved
system (2.13) has a bounded solution ~(x; e) w h i r
is close to ~(x; 0) and connects the critical points
Moreover we may assume that ~(0; e) = 0 and tha
Y(x; e) 4:0 for x 4: 0. In order to prove uniquenes
suppose by contrary that there is a sequence e,
and corresponding non-trivial bounded solutio~
ft,(x) and ~ ' ( x ) of (2.13) so that ~" is not a shil
of ~n. Since the above solutions are uniforml
bounded, without loss we may assume that bot
sequences £, and ~" converge uniformly on finil
intervals to the solution ~(x; 0). As it follows fro~
lemma 2.2 below, fo r sufficiently large n the soh
tions ~, and ~" connect the critical points ~L a~
~a and thus the corresponding unstable and stab
manifolds intersect along two close trajectori~
D. Michelson / Steady solutions of the Kuramoto-Sivashinsky equation
93
This, however, contradicts the transversality of the
intersection for e = 0. Thus it remains to prove
- o o ) . Then + t o ( - o o ) is an accumulation point
of the zeros of yl(x).
Lemma 2.2. Let ~ , ( x ) be a sequence of bounded
solutions of system (2.13). Suppose that ~ ( x )
converges uniformly on bounded intervals to
~(x; 0) as e = e, ~ 0. Then for sufficiently large n,
lim~ ~ _ ~ ~ ( x ) = ~L = (1,0,0), lim~ _,~ ~ ( x ) =
~R = (-- 1,0,0).
Proof. Indeed, otherwise y l ( x ) is of a constant
sign for x greater than some x 0. The function
L(f~) =y2 + y2 + y2y 2 _ 2c2y2 is then a Liapunov
function of (2.1) for x > x 0 since d L ( ~ ) / d x =
2yly~, and therefore ~ ( x ) has a limit at + to.
Proof. First observe that for any neighborhood
UR of ~R which is disjoint with ZL there exist
e o > 0 and a neighborhood Ul{ c Up. of zv. such
that for lel < eo any bounded solution of (2.13)
which belongs to U~ for some x = x o will stay in
U R for x > x o. Indeed, otherwise there exists sequence e,--* 0 and corresponding sequence of
b o u n d e d solutions [ , ( x ) so that ~ , ( x 0 ) ~ R,
~ ( x n ) ~ U R for some x , > x o and ~ , ( x ) ~ U R for
x o <_x < x , . By uniform boundedness of ~ ( x ) we
m a y assume that [ . ( x ~ ) converges to ~, ~ R 3 \ U R.
Clearly ~, belongs to the bounded trajectory
~(x; 0). N o t e that x~ -~ o¢ since otherwise ~, would
be connected with ~R in backward direction by
~(x; 0). Now, leaving ~, by the trajectory 2(x; 0)
in backward direction we should reach in a finite
time T a small neighborhood U L of zL- By continuity, for large n also ~,(x~ - T ) ~ U L. However
x~ - T > x o for large n and therefore ~ ( x ~ - T )
U~. Hence a contradiction. Now, for sufficiently
small UR and e any solution of (2.13) which stays
in U R in forward time will tend to ~a. We choose a
corresponding U~ and a small e0, and similarly for
the point 2L- Select T such that ~ ( - T ; 0 ) ~ U[
and 2 ( T ; 0 ) ~ U~. Then for sufficiently large n,
2,( - T ) and ~n(T) belong to the above neighborhoods and thus ~ connects the critical points.
Q.E.D.
3. Periodic and quasi-periodic solutions for c << 1.
Consider eq. (1.5) for 0 < c << 1. As in [9] we are
looking for periodic solutions with frequency ~ =
1 + 0 ( c ) . It is convenient to rescale the variables
so that the period is independent of c. Introduce
= WX,
Then
_ _
d3z
d~ 3 +
z 2
d z = e2 _ __
)k-~---~
2 '
Lemma 2.3. Let .~(x) be a bounded solution of
(2.1) such that y ( x ) has no limit as x ~ + oo(x
(3.2)
where
e = c / w 3,
X=w -2=l+t~(e).
(3.3)
Periodic solution of (3.2) with period 2~r could be
found by means of a power expansion in e
Zper =
~ Z.(tZ)en, ~= 1 + ~ Xne".
n=l
(3.4)
n=l
Substitution of (3.4) into (3.2) gives
z['" + z{ = 0,
z ; " + z~ = 1 - XlZ ~ -- 2?//2,
z ; " ' + z~= - h x z ~ - h a z ~ - z l z 2,
(3.5)
etc.
Thus z I b i t -t- all sin ~ + b u cos ~. Shifting ~ one
m a y always assume that b l l --- 0. In order to avoid
resonance in the equation for z 2 one should assume that hi = 0, bit a l l -----0 and b2o/2 + a21/4 =
1. Here there are two possibilities. If a u - - 0 then
b i t = 5: v ~ and we recover the stationary solution
z - - 5 : e v ~ . If b i t - - 0 , then a x l = 5:2 and as it
=
We conclude this section with the following
observation:
(3.1)
7. = y / / w 3.
D. Michelson / Steadv solutions of the Kuramoto-Sivashinsky equation
94
follows from consequent equations,
z , ( ~ ) = ~ a,k sin kS,
equation i~(z,/~, e ) = 0 could be solved as
(3.6)
Pz-=~= ~ a , s i n , ~ +
k=l
~b, cosn~=f(b0, al,t,
n=2
so t h a t Z p e r ( ~ ) is an odd function. The solutions
with a n = + 2 an a n = - 2 are related by the shift
--, ~ + ~r. Let us select aal = - 2 so that for small
e > 0, z'(0) < 0. The first two terms of the expansion are
Zper = - 2esin ( - {e 2 sin2( + 0 ( t 3 ) ,
n=2
= f ~ ( b 0, a,, ~) +fb(b0, aa,/t),
where f~ represents the first sum in 2 and fb the
second one. Note that b0, a x and /, enter F only
through quadratic terms. Therefore
d f ( 0 ) = 0.
(3.7)
)t = 1 + e2/12 + O(e4).
(3.11)
(3.12)
Next, the m a p / ~ vanishes for all constant z = bo.
Therefore
Actually all X2,+1 = 0 and a,k = 0 if n - k is odd.
The expansion in (3.4) could be justified rigorously
using the L i a p u n o w S h m i d t reduction. Namely, a
periodic solution of (3.2) with period 2~r may be
expanded as
f(bo,O, #) = 0.
zp¢~(~) = b o + a 1 sin
In view of (3.13) and (3.14) and the analyticity of.
+
a, sinn~+
b, cos n( . (3.8)
n~2
Finally, F(z(~), p.t, ~) = F ( - z ( - ~ ) ,
hence for b0 = 0
fh(O, al,/z) -- O.
2+z2/2
(3.9)
IX, e), and
(3.141
f~( bo, al, ~ ) = a l f ' ( bo, a 1, p.t),
fb( bo, al, I~) = a~bof~( bo, al, bt),
(Shifting ~ if necessary we may assume that there
is no cos ~ in the expansion.) Let H be the Hilbert
space of functions as in (3.8) with scalar product
(u, v) = fo2"(u '"~'" + u6)dx and H 1 = L2[0,21r ].
We consider the mapping
F:(z,/x,e)~z'"+(l+lx)z'-e
(3.13)
(3.15
where the maps fd and fh' are analytic. By (3.12
also
f ' ( 0 ) -- 0.
(3.16
Now consider the remaining equations ( I - P1)°~
= 0. They are
oo
as a mapping from H (9 C (9 C into H v Clearly F
is differentiable and even analytic. Denote b y / f c
H 1 and /-)1 c H I the subspaces spanned by sin n(,
cos n~, n ~ 2 and let P: H ~ / 4 , PI: H1 ~/~1 be
the corresponding orthogonal projectors. Consider
the map
pa I - l ~ a.an+l
F=PloF:
for sin ~ component and
H~C(gC~/)I.
(3.10)
N o t e that the differential dF(0) at zero when
restricted to the subspace /-) is an isomorphism.
Thus the implicit map theorem applies and the
~ ~.. b,,bn+l=O
n=l
(3.1~
n=2
for cos ~ component,
OQ
--boal-- I ~.. ( a , + l - a , - 1 ) b , = O
(3.11
n=2
-e2+¼
E a ,2+ ¼
n=l
b~+ ~b
1 o2 _- 0
n=2
for the constant component.
(3.1!
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
If a 1 = 0, by (3.13) and (3.19) we obtain the
trivial solution z = eV~-. Now let aa be small and
different from 0. We will show that b o = 0. Indeed,
otherwise divide (3.18) by boa l. Then
' 1 - a n _' 1 b
an+
1 + ~a 1
=0,
(3.20)
I.n-2
where a~ = a,,/a 1, b~, = bo/alb o. In view of (3.15)
the expression in the brackets in (3.20) is an analytic function of a 1, b 0 and ix. However, a~ is
small and thus (3.20) is impossible. It then follows
from (3.14) that all b, = 0. Then from (3.17) we
recover
IX = ½a 1 ~ a',,a'+1 = alf~(al, ix),
(3.21)
n=l
where f~ is an analytic function. By the implicit
function theorem (3.21) could be solved for #:
IX= a l % ( a l ) .
(3.22)
Finally, eq. (3.19) is used to compute a 1,
(3.23)
Since by (3.16) E,_2(a,)~
, 2= O(a~) we get for
small e
a 1 = + 2 e + 0 ( a 3)
(3.24)
and then by (3.22) and (3.11) recover z and 3` = 1
+ IX as analytic functions of e.
Using a computer, we have calculated the first
100 terms of the expansion in (3.4). The radius of
convergence is
[el < R - - 3.558.
By (3.3) one can also reconstruct the values of to
and c corresponding to e. The to, c curve is shown
on fig. 1. The maximal c = cm~,~ 1.266 corresponds to to = 0.84. The frequency too = v~-/2, as
mentioned in the Introduction, is maximally
95
amplified by the linear terms in (1.1). The corresponding value of C2 = 1.17 in the graph is close to
the mean propagation velocity c02-- 1.2 of a turbulent flame as calculated by the numerical experiment in [6]. The portion of the to, c curve on fig. 1
extending from Po through P1, P2, P3 to Q corresponds to the above domain of convergence [eJ <
R. For e close to R one cannot rely any more on
computations based on 100 terms of the expansion
in (3.4). To circumvent this difficulty, we approximate (1.5) by a difference scheme in (4.1) and
compute instead the periodic solutions of the
scheme. The entire graph on fig. 1 is actually based
on such computation with a step size A x in (4.1)
being l / N where l = l ( c ) is the period of the
solution and N = 120 is the number of grid points
in the period. The periodic solution is found by
following the fixed point of a corresponding
Poincar6 map, while the fixed points are computed
using Newton's method. One should note that an
expansion as in (3.4) exists also for the periodic
solution of the difference scheme. We found that
for N = 120 the periodic solutions for the difference and differential equation in the domain
PoQ differ by less than 0.5%. The points Pi, P/ on
the graph are parabolic points, i.e. the eigenvaiues
of the Jacobian of the Poincar6 map at these
points are 3 ` 1 = 3 ` 2 = - 1
or 3 ` 1 = 3 ` 2 = 1 - At
P1, P2,Ps,P6,Ps',Pg the eigenvalues are 3`x=3`2 =
- 1 while at P3, P4, P~ and P~ they are 3,1 = 3`2 = 1.
In all cases the Jacobian has a single eigenvector.
At Pv the periodic solution coincides as a double
loop with the one at P1 and follows (as a double
loop) the branch P1P0 until the endpoint Ps! Thus,
for the same values of e the frequency to at the
branch PvP8 is half the one at the branch P1P0. In
the neighborhood of P4 we were surprised to find
another fixed point of the Poincar6 map corresponding to non-odd periodic solution. This solution branches out at P4 and continues through the
points P~,P[,P~,P~. By symmetricity, - y ( - x )
would be another non-odd solution. The values of
to and c at the parabolic points appear on table I.
Our computations show that the variable e = c/to 3
grows monotonically as one moves along the to, c
D. Michelson / Steady solutions of the Kuramoto- Sivashinsky equation
96
Table I
The values of c and w at the parabolic points
Parab. points
P1
P2
P3
P4
P5
P6
P7
P~
P[
P~
P~
c
w
Eigenvalues
0.3195
0.9961
- 1
1.2660
0.8424
- 1
1.2664
0.8372
I
0.5982
0.5548
1
0.5796
0.5502
- I
0.3407
0.5012
- I
0.3195
0.4981
1
0.606
0.554
- 1
0.9165
0.479
- 1
0.917
0.477
1
0.8365
0.393
1
curve from Po towards Q and reaches its maximum
at the point Q. Another experimental observation
is that the coefficients h , of the expansion in (3.4)
are positive. Hence at Q, e equals the radius of
convergence R and d h / d e is infinite there. We
need the above arguments since a direct computation of the singular point of h(e) using, a reasonable number of terms in (3.4) is too inaccurate.
Our next goal is to study the Poincar6 map
associated with the periodic solution Zper in (3.7).
Change in (3.2) the variable
z ---."z / e
(3.25)
and rewrite the corresponding equation as a system
satisfies the identity
j ~ = ~ - 1j,
(3.27)
where J = j - l : R 2 ~ R 2 maps the pair (z, z') into
( - z , z'). Consider the differential d ~ ( ~ 0 ) of the
map ~ at 2o. Clearly,
I d e t d ~ ( ~ 0 ) I = 1.
(3.28)
In order to compute d~(~0) one should solve the
linearized equation
z " + •(e)z'=
-eZperZ.
Expand
z = z 1 + ez 2 + d~(e2 ).
d~
d--'-~ = f ( z ' e) = ( z', z", e - ez2/2 - X( e)z'),
Z= (Z
, Z', z " ) .
(3.29)
(3.30'
Then,
(3.26)
The plane z " = 0 intersects the periodic trajectory
zper at least at two points ~o = (0, z~r(0),0) and
zl = (0, Z~r(~r),0 ). (We use the old notation zp~
for the rescaled Zp~r/e ). Denote by R 2 the plane
z ' = 0, by D 2 c R 2 a small disk centered at ~0
and by ~ : D 2 ~ R 2 the corresponding Poincar6
map. Observe that the flow in (3.26) is volume
preserving since div f(~, e) = 0. Hence the map
preserves the measure (e - ez2/2 - ~ ( e ) z ' ) d z dz'.
r
< 0 this measure is positive in a
As far as Zp~(0)
neighborhood of Zo. For small e, z ~ ( 0 ) = - 2 +
O(e) < 0. Our computations show that z~¢~(0) remains negative along the whole curve on fig. 1.
Besides the volume preservation, the flow in (3.26)
is invariant under the change of variables
(z, z', z") ---, ( - z , z', - z"), ~ ~ - ~ . As a result
Zl'"+z{=0,
zff'+z~=2sin~.z,,
(3.31
so that
z = a ~ ( 1 - e, s i n , ) + a 2 ( s i n , + e, + e ~ )
cos 2~ \
+a3(cos~+e~]+O(e2).
(3.37
Now we impose conditions on a 3 and ~ = 2rr +
so that
"(0) = O,
z;~,(2~r + z ~ ) + z"(27r + z ~ ) = 0(A~2).
(3.3:
Thus
a3 = - - 2 a l e + ~ ( e 2 )
(3.3
97
D. Michelson/ Steady solutionsof the Kuramoto-Sivashinsky equation
and
A~ = - z"(2~r ) / ( Zp'/ (2~r ) + z'"(2~r))= O(e2).
(3.35)
The transformation d~(~0) is defined by
i.e. the higher differentials with respect to 8z and
8z' are of order 0(62), and
d ~ ( Sz, 8z'; e)
= [/+2~re(O 1
d ~ ( e o ) : (z(O), z'(O))
1)+O(e2)](fizZ,),
(3.41)
d2,~(Sz, 82'; e)
(z(2~r) + A{z~r(0 ), z'(2~r) + A{ z;'[(0)).
(3.36)
j
(3.42)
An easy computation shows that
d.~(~o) = / + 2,re( 01
1) + 0(e2).
(3.37)
1
8z = ~ ( . -
For small 6 the eigenvalues of d~(~o) are
h L 2 = e +-'"°<~), ao(e)=2~re+d)(e2),
(3.38)
i.e. they are complex conjugate and on the unit
circle. Thus for small e the periodic solution is
elliptic. This domain of ellipticity extends until the
point P1 (see fig. 1). The other elliptic domains are
P2P3, P4Ps, PrP7 and P7P8 (the last one coincides
with PoPD. At the bifurcating branch of the nonodd periodic solution the elliptic domains are P4P~
and PIP4. In the elliptic regions one would expect
the existence of nested invariant tori surrounding
the periodic orbit. In order to prove it rigorously,
one should verify the conditions of Moser's twist
map theorem (e.g. see [13], pp. 225-228). First
recall that the map ~ depends analytically on z
and z' in a small neighborhood D E and preserves
the positive measure ( 6 - ~ ( 6 ) z ' - 6 z 2 / 2 ) d z d z '.
Next we should compute the Birkhoff normal form
of ~ (see [13], pp. 158-159). For small 6 this
could be done analytically. Change the variables
~Z=Z,
~Zt=Zt--Z;er(O).
Then a quadratic change of variables (see the
appendix)
~) + ¢(6),
(3.43)
8 z ' = { ( u + f i ) - ~6(u 2 - 6ufi + ~2) + O(e)
brings ~ to the form
~ : (U, U) ~ (Ul, Ux),
ul = ei'u + ez¢(lul4),
(3.44)
where
O~ = 0/0(6 ) + O/1(6) UU ,
3rr
Oil(E) ~- -- -~-E+¢(E2).
0/0(6 ) = 2q7"6 + 0(62),
(3.45)
Thus for small ~ > 0
al(e ) • 0
and
nao(e )*O(mod2~r),
for 1 <n_< 4,
(3.46)
and the conditions of the twist map theorem have
been verified. The result implied by the theorem is
as follows:
(3.39)
In the new variables the Poincar6 map ~ depends
analytically on 8z, 8z' and ~ in a neighborhood of
0. Our computations (see the appendix) show that
~ ( 8z, 8z'; e) = d ~ ( S z , 8z'; e)
+ d 2 ~ ( S z , 8z'; e) + O(e2), (3.40)
There exist numbers ro > 0 and eo > 0 such that
for any e ~ (0, %) and any o~ ~ (a0(e), ao(e ) +
al(e)r 2) satisfying infinitely many inequalities
[ ~o
Pq I >- flq-"
(3.47)
with some positioe fl, • and all integers q > 0, p,
98
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
there exists an invariant curve S of ~ of the form
S: 8 z = S z ( o p ) ,
3z'=dz'(ep),
z"=O,
q ~ R.
(3.48)
Then for small [171 and ~A/', 0 is defined uniquely
and satisfies the above conditions. Now we introduce on 3 - global analytic coordinates (% #), qo
[0,2~r], #~[0,2~r] such that the trajectories of
(3.26) on 9- are described by
The functions 8z(~p) and ~z'(~p) are periodic with
period 2rr and depend analytically on % The curve
in (3.48) is a perturbation of order re of a circle
tin = % + (~o/2w)va(mod2w).
8z = - r sinqo,
Along a trajectory, parameters ~ and # are related
by the equation
8z' = rcosq~,
(3.49)
r=[-~-( 1-~]'/~2~r17
]]
d~ = g(O, ¢p; 17),
d---~
and the map induced by D~ on the curve is
¢o --,
+ 0,.
(3.5o)
The solutions of (3.26) which originate at the
curve (3.48) form a two-dimensional torus ~Y" and
for - oo < ~ < oo each trajectory on the torus is
everywhere dense in 27". We shall show that the
above solutions are quasi-periodic functions of ~.
Let .A/" be a small tubular neighborhood of the
periodic solution ~ ( ~ ; 17). One can parametrize
~V" in the longitudinal direction by a parameter
~ [0, 2 ~r] so that
a) e i# is an analytic function of ~ V " and
17~ [ - 170, 17o];
b) # = - 0 (mod 2~r) at the section z ' - - 0 in a
neighborhood of z0;
C) ~('~'per(~; 17))= ~ ~ [0,2~r].
To do this, one could for example define a reper
drp = to/2~r,
~-~
e2(~; 17) = d2~p~(~; 17)/d( ~,
~ = fl0# + h0(v~),
(3.53)
where /30 = (2~r)-2f02~f02"g(v~, % 17)dO drp and
h o ( # ) is a quasi-periodic function of the class
Q(1, ~0/2~r) (see [13], pp. 258-264). Note that for
small ,A/" and 17, g(O, cp; 17)--1 and d ~ / d # is
bounded away from 0. Thus the inverse function is
= flol~ + hl(~ ),
(3.54)
where hl(~ ) is quasi-periodic and belongs to the
class Q(flo 1, flo ko/2 ~r). Since ~ is a periodic function of % #, the trajectory ~(~) is a quasi-periodic
function of ~ of the class Q(flo 1, flol~/2~r).
Finally we will show that the integral
(3.55)
"r=0
e 3 = e~ X e~.
For small 17 these vectors are independent. If ~(0)
is a unit vector normal to the plane z " = 0 and
fi(O) = ~,ai(17)ei(O; 17), define
(3.52)
where g(O, % 17) is analytic in ~, ¢p and 17 and
periodic in ~, ~0 with period 2~r. In view of the
conditions in (3.47), ~ could be written as
el(,~; 17) = d£pe~(~; 17)/d~,
is quasi-periodic too.Consider the invariant curve
in (3.48). We may assume that 8z(0)= 0. As il
follows from (3.27), the curve
(-
17)=
(3.51)
(3.561
17).
N o w let ~ satisfy (c) and be constant along the
sections
is also invariant under ~ . By uniqueness ~z(no~
= - S z ( - n t o ) and 8z'(nt0)= 8 z ' ( - m o ) for all J
Z. Hence
(Zr--,~per(~; E))'n(~; 17)~-~0.
~Z(qO)= --~Z(--Cp),
~Z'(lp)'~-~Zt(--~).
(3.57
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
Let ,~(~) be a trajectory of (3.26) which originates
(when ~ = 0) at the point (~z(qg), 6z'(q0), z" = O)
and let ~'(q~) be the value of ~ as the trajectory
reaches the point (8z(q~ + to), 8z'(q~ + to), z " = 0).
Denote g(ep)= f~<~)z(~)d~. Clearly, g(q~) is an
analytic function of tp with period 2~r. Observe
that ( - z ( - ~ ) , z ' ( - ~ ) , - z " ( - ~ ) )
is a trajectory
of (3.26) which connects the points ( 8 z ( - q o to), 8z'(-q~ to), z " = 0) and (Sz(-q~), 6z'(-~p),
z " = 0) as ~ changes from - T ( ~ ) to 0. Hence
99
rational number) lies in the range (ao(e), ao(e ) +
a~(e)r~). It is quite possible that for some periodic
solutions the average in (3.60) is non-zero and the
corresponding integral in (3.55) has a non-zero
slope.
4. Numerical experiments
-
g(-~o-w)
= fo
-z(-~)d~=
-,r(~o)
-g(~o),
and consequently
fo2~g(qo) dep = O.
(3.58)
In order to gather more information about the
set of bounded solutions of (1.5), especially for
intermediate values of c, we have approximated
(1.5) by a difference equation and solved it on a
computer. The difference scheme employed was
(Yj+3 - 3Yj÷2 + 3Yj+I - Y j ) / A x 3
+ ( yj+ 2 - Yj+ x)/Ax = c 2 - ~( Y?+2 + Y?~ x),
j ~ Z,
As a result of (3.58) and (3.47) the sums
N
E g(~P + nto)
(3.59)
n~0
are uniformly bounded. This implies that the average
lim
z(,r) d'r = O.
(3.60)
Hence w(~) is a quasi-periodic function of the
class Q(flo 1, flolw/2rr). Recall that the function
w(~) is proportional to v(x) in (1.3). Thus, we
have proved the existence of slowly propagating
quasi-periodic flame fronts (in Sivashinsky's
model). These waves have the basic frequency of
order 1, and a modulation with an incommensurable frequency of order w/2~r.
Moser's twist map theorem tells us that the
relative measure of the invariant tori in ,/~ tends
to 1 as ,W" shrinks to the periodic orbit. The
domains bounded by each pair of (eoelecial) tori is
invariant under the flow. Besides the quasi-periodic solutions there is also an infinite set of periodic solutions which is everywhere dense in , ~ .
The frequency w of periodic solutions (which is a
(4.1)
where yj is the value of the grid function at
xj =jAx. The scheme in (4.1) maintains the symmetry of the equation in (1.5) in the sense that it is
invariant under the transformation
j+-j,
y+-y.
(4.2)
We have investigated mainly the odd solutions of
(4.1), i.e. those which satisfy the initial conditions
Y0 = 0 ,
Yl=-Y-1
=Ax's,
(4.3)
where s is a parameter. The values of yj, j > 2 are
then calculated by (4.1) until lyjl exceed a certain
large number Ym~. Denote by xm~(s) the point
xj = j A x where for the first time lyjl -Ymax" Recall
that for c E [0, Co] all bounded solutions of (1.5)
lay in a strip [y[ < Ym~ where Ym= depends only
on c 0. The same is true for eq. (4.1). Thus for
bounded solutions Xm~,(s ) = oo. It was a surprising empirical observation that with a few exceptions all local maxima of Xma~(S) have been
infinite. Hence a sequence s~ < s 2 < s 3 with nonmonotone Xm=(Si) would imply that for some
s ~ Is 1, s3] the corresponding solution is bounded.
Of course one cannot be sure that there is only one
bounded solution. Hence the interval was subdi-
100
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
vided into, say, 100 subintervals, and if no additional m a x i m a appeared, it was assumed that
there is only one bounded solution. Then a Golden
Section method for a univalent function was employed in order to converge to above s. As suggested by the theory in sections 2 and 3, only
solutions connecting the critical points were isolated. Since these solutions are asymptotically unstable, in order to reach Xm~x ~ 100 the double
precision on C D C (i.e. 30 decimal digits) was
required. W e set Ymax= 10, AX ~ 0.05 and considered c in the interval [0,4]. The results are as
follows. For c > c 1 -- 1.283 problem (4.1), (4.3) has
a single bounded solution. This solution corresponds to s < 0, vanishes only at j = 0 and tends
to the critical points. We followed this solution
until c = 0.2. As c decreased, the above solution
did not change its shape, i.e. vanished only at
j = 0, until we reached c - - 0 . 3 . Then additional
zero point evolved which split in two as c decreased below 0.3. At c = 0.2 our solution was
almost unseparable from an invariant tori. At c =
c~ = 1.283 a bifurcation occurs. A new bounded
solution y)X) with a slope s 1 > 0 is born. This
solution has exactly one zero for j > 0 (i.e. one
change of sign) and connects the critical points.
F o r c < c~, y)l) splits into two similar solutions
y)l) and yj-1) with slopes s 1 and s_ v At c = c z ~"
1.274 a second bifurcation occurs where another
solution y)2) with a slope s 2 < 0 is formed. For
c < C2, #2) splits into yj2) and y)-2) with slopes
s2, s 2 < 0 . Both y)Z) and y)-2) have exactly 2
zeros in the domain j < 0. As c decreases, at
c = e 3 -- 1.2679 and c = c 4 --- 1.2673 solutions y)3)
and y)4) are formed with correspondingly 3 and 4
zeros in the half line j < 0. On the other hand eq.
(4.1) like (1.5) has periodic solution. Namely, for
A x ( 0 ) = 2,r/N, N integer, there exist analytic
functions Ax = Ax (e), c = c(e) and periodic grid
function yfper)(e), j ~ l with y(per)(e) =y}+P%)(e)
which satisfy (4.1). We have computed, the above
functions for N = 120. The graph of c(e) versus
the discrete frequency ~o(e)= 2~r/(NAx (e)) is almost the same as for the periodic solution of (1.5).
In particular, the maximal value C~x of c(e) is
1.2664 instead of 1.2662 for the differential problem. Our computations at c --- Cm~x with the corresponding Ax = Ax (e) --- 0.0625 show that the set
of odd bounded solutions of (4.1) consists of two
sequences ( y ) " ) } and ( y ) - " ) } , n = 0 , 1 , 2 , . . . with
slopes s, and s n and a periodic solution yj(r~o.
The functions y)n) and y ) - " ) have n zeros (i.e.
changes of sign) in the domain j > 0, and tend to
the critical points +Cma~'v~-. The slope s, is
positive for n odd and negative for n even. The
sequences s2, and sz,+a, n > 0 are decreasing,
S2n, S2n_ 1 are increasing. The limits Sev =
lim S2n
lim s _ 2 , a n d Sod = lim s2,+1 =
lim s _ 2 , _ a are the slopes of the periodic solution
y(per) corresponding to Cm~x at j = 0 and at j =
N/2. The functions y)2,) thus tend to yjpeo while
y)2,+1) tend to ySP+~/2. The above statement is
indeed a conjecture which is based on actual computation of yS-+") for Inl < 2 0 and x i < 8 0 - 9 0 .
On an interval 0 < x < 80-90 we observe about
20-25 local extrema of the function y)"). Note
that the condition Xm~x(S)> 90 sometimes determines s up to 25 significant digits! Based on the
above results we also conjecture that there is
sequence of bifurcation points c, which tend,
monotonically to Cm~,. At e = c, the solution y~"
is b o r n and splits into y)") and y}-") as c de
creases beyond c,. On fig. 2 the solutions y~0)
#1), # - , ) , # 3 ) a n d # p e r ) a t C=Cm~x are dis
played. We followed y i(°) until c = 0.2 and y j - 2
and y)-4) until c = 0.3 (see figs. 10 and 11). Th,
solution yj-4) disappeared somewhere betweel
c = 0.3 and c --- 0.295 while yj-2) disappeared be
tween c --- 0.295 and c = 0.293. We conjecture tha
each solution y)") exists until some c~. At c =
c~, yJ") becomes a limit of a sequence of bounde,
oscillating solutions and disappears for c < c~.
As c decreases beyond Cm~x -- 1.26644, there is
sudden change in the set of bounded solution. Tlz
periodic solution splits into two. The elliptic one:
surrounded by a thin invariant torus. As c d l
creases from c ~ to c -- 1.26603 at the parabol
point Pz (fig- 1), the thickness of the maximal tort
measured b y the slope s first increases from 0 1
2 x 10-3 and then decreases back to 0. By the~
=
D. Michelson / Steady solutions of the Kuramoto-Sivashinsky equation
101
4--
~-
O-
-2-
a
-4
5
lO
~5
20
25
X
2-1 2
l-
0-I0
3
-::'0 -
2
~_ - t -
-2-30 -
-3*
b
-40
, j r ~ i ~ ~ , t i ~ , , , i , , , , i ,
5
t0
:15
20
X
2
c
1
-4
-3
,,~,1,,,,i,~,,i,~,,i,,,,1~,,~
-2
-!
0
t
2
I
3
Y
Fig. 2(a) c = Cmax --- 1.266, a s y m p t o t i c s o l u t i o n s yt0), y O ) , y ( - 1 ) a n d y O ) (b) c = Cmax, c u r v e s 1, 2, 3 r e p r e s e n t t h e B u n s e n f l a m e s
c o r r e s p o n d i n g to t h e s o l u t i o n s y(0), y O ) , y(3). (c) c = Cmax, c u r v e s 1 a n d 2 c o r r e s p o n d to t h e s o l u t i o n s Y p e r a n d y(3).
rem 2.1 the hyperbolic periodic solution may not
be isolated f r o m other bounded solutions. Denote
b y I the set of slopes s corresponding to the
b o u n d e d odd solutions of (4.1). At c = 1.2663 (i.e.
between the points P3 and P2) the set I is as
follows. First, I splits into two parts I 1 and I2,
where I 1 is concentrated near s = -3.0275 and I 2
near s = 1.36. These are approximately the slopes
of periodic solutions at ~ = 0 and ~ = rr. The lower
b o u n d of 11 is s o = - 3 . 1 5 6 7 , the upper bound is
s 2 - - - 3 . 0 1 1 1 . These slopes correspond to solutions y(0) and y(2) above. The slopes s of odd
solutions which lie in the invariant torus form
two intervals ,/1 c 11 and J2 c 12, where .]1 -[ - 3.0266, - 3.0247]. The set I x \ I n t ( J x ) consists
of a Cantor type set K 1 and a discrete set D 1. The
solutions corresponding to s ~ K 1 are oscillating
and p r e s u m a b l y nonquasiperiodic (we computed
a b o u t 20 oscillations). The solutions corresponding
to s ~ D 1 are asymptotic (i.e. connect the critical
points) and are isolated. The set of the limit points
of D 1 is exactly K 1. Thus, all oscillating (odd)
solutions f r o m K 1 are limits of asymptotic solution with increasing numbers of zero. The set 12
has a similar structure. Of course our claim is a
conjecture based on numerical experiments. We
are scanning a certain interval of slopes s with an
increment As and are looking for local maxima of
Xmax(S ). Suppose they a r e s 1, s 2 , . . . , s,. If A s is
sufficiently small we usually did not " j u m p over"
such maxima. Then for each s i the neighborhood
(s i - A s , s i + A s ) is scanned with smaller As1, say
102
D. Michelson/ Steady solutions of the Kurarnoto-Sivashinsky equation
As 1 = A s / 4 0 0 . If no additional maxima were
found, we concluded that there is a unique maxima in the above interval which corresponds to an
asymptotic solution. Additional scannings and
convergence to the maximum always supported
this conclusion. If, however, new maxima were
found, their distributions on a smaller scale repeated the distribution of s 1, s2,..., s,.
The picture described above does not change
qualitatively until the invariant torus disappears as
c reaches the point I72. For values of c between P2
and P4 both odd periodic solutions are hyperbolic.
Here the set I consists of a Cantor type set K and
a discrete set D of isolated asymptotic solutions.
In general we did not investigate the non-odd
bounded solution. However, the presence of nonodd periodic solutions of the branch PsP#P~
considerably effects the shape of odd bounded
solutions. For c between P# and P~, i.e. 0.83645 <
c < 0.91695 the picture is the most interesting. On
fig. 3 one can see the projections of 4 periodic
orbits on the y, y ' plane for c = 0.85. The reflection y ~ - y , y ' ~ y ' would provide two more
non-odd periodic solutions. On figs. 5a-5d we
have displayed a typical odd bounded solution for
c = 0.85 (x > 0). The corresponding flame front
v ( x ) on fig. 5d looks completely chaotic. Indeed,
numerical experiments with the P.D.E. (1.1) in [6]
resulted in flame fronts of the above type. Moreover a closer look on fig. 5a reveals that the
trajectory follows closely the periodic orbits for
the same value of c. Indeed, three loops around
the critical point y = - ~ = 1.2, y ' = 0 are very
close to the periodic solutions 3 and 4 on fig. 3 and
to the one on fig. 4d (corresponding to the point
Pff). The large cycle lies near the odd periodic
solution # 1 on fig. 3 and four loops around the
right critical point follow closely the reflected nonodd period solutions. From fig. 5c one can understand the order in which the loops are followed.
The small oscillations near - 1 correspond to the
left centered loops, the one near 1 - to the fight
centered ones and the simple wave at 88 < x < 95
corresponds to the large symmetric cycle. On fig.
6a we observe another odd bounded solution for
2-l
O"
y'
-I-
-2"
-3
,
-3
~
,
,
i
-2
,
i
,
,
i
,
,
,
-I
~
i
0
~
t
,
,
i
~
t
,
,
,
i
2
Y
Fig. 3. Periodic solutions at c=0.85: 1-odd solution (right
branch of the ~ - c curve), 2-odd solution (left branch), 3-nonsymmetric solution (right branch), 4-nonsymmetric solution
(left branch).
c = 0.8. Observe that the PJP~ curve on fig. 1
crosses the level c = 0.8 only once. Hence there is
only one non-odd period solution and its reflection, and as a result the trajectory is more trivial.
The odd period solution # 1 is replaced here by
# 2 . On fig. 6b an odd asymptotic solution (for
x > 0) is displayed. This solution is almost undistinguishable from the one on fig. 6a (their initial
slopes differ by 3 x 10-14!) until it starts to spiral
around the critical point. As we already mentioned, in the hyperbolic domains there is a Cantor
type set of slopes corresponding to odd oscillating
bounded" solutions. These bounded solutions follow different combinations of the few periodic
orbits. In order to understand which combination.,
are possible one should study the Poincar6 map #
acting in the plane y " = 0. The periodic orbit.,
correspond to the fixed points of ~ . In the hyper.
bolic case, there are one dimensional stable an(
unstable manifolds originating at these points. Ap
parently these manifolds intersect at heterocliniq
or homoclinic points. In the last case it is wel
known (e.g. see [15]) that there exists an invarian
Cantor set in the plane y " = 0 so that ~ acts on i
as a Bernoulli shift.
We mentioned in the introduction that th
branch P4P~ on fig. 1 actually continues beyon
the point P~ and terminates as ~ ~ 0 and c -
D. Michelson/ Steady solutions of the Kuramoto-Sioashinsky equation
2-
103
E-
O-
-2"
-2
b
a
,
-3
,
,
,
i
-3
,
,
,
]
i
-2
,
,
,
,
i
-~
,
,
,
,
i
0
,
,
,
,
-3
i
~
,
2
i
,
,
i
-3
[ , +
,
i
-2
,
,
,
,
i
-i
,
,
,
,
i
0
Y
,
,
,
,
I
i
2
Y
2--
2-
O-
O-
-i-
-2-
-2-
d
C
-~
'
-3
'
'
'
I
-2
'
'
'
'
I
'
'
'
-i
'
I
0
'
'
'
'
I
1
'
'
'
'
I
2
-3.
,
,
,
'
,
-3
'
'
-2
'
/
'
'
'
-t
Y
'
I
0
'
'
'
'
I
t
'
'
'
'
1
2
Y
10
-40
,
0
,
,
,
i
iO
,
,
,
,
i
20
,
,
,
,
i
30
,
,
,
,
i
40
,
iii
I
50
X
Fig. 4(a) Soliton for c = i n f C s ~-0.835.(b) Soliton for c=0.848. (c) Soliton for c = sup Cs =0.86. (d) Non-symmetric periodic
solution at the point P~(c = 0.866). (e) The flame fronts 1, 2, 3, 4 correspond to figs. 4c, 4a, 4d, 4b.
104
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
2~
2--
t1-
O-
">
-i-
-2-
-S
'
'
-3
'
'
I
-2
'
'
'
'
(''
'
'
;
1 '
0
-!
'
'
'
i
'
'
'
'
i
I
2
'
'
'
'
I
3
-2
-3
' ' ' '
I ' ' ; ' 1 ' ' ' ' 1
-2
-t
Fig. 5a. One odd "chaotic" solution for.c = 0.85 (only the part
of trajectory for x >_0 is shown).
3~
2
t
Fig. 5b. The same trajectory as in fig. 5a projected on the y'y"
plane.
tO--
2-
j
t-
>..
' ' ' ' 1 ' ' ' ; I
0
y.
Y
9-
-tO-
-20"
-2-
-3
"
I
;~0
'
I
40
'
I
80
'
I
80
'
I
tO0
'
I
i20
-30
r~
140
×
Fig. 5c. The same as in fig. 5a in the x, y plane.
---0.86 at the "soliton" displayed on fig. 4c (the
n a m e " s o l i t o n " is used because y ( x ) tends to the
same limit cx/2- as x ~ _+ o0). N o t e that the eigenvalues hi, h2 of the Jacobian d ~ at the periodic
orbit tend at the same time to oo and 0. Because of
it we could n o t reach b e y o n d the point P~. One
s h o u l d m e n t i o n here an important work of
T z v e l o d u b [8] which only recently came to our
attention. U s i n g an entirely different m e t h o d he
o b t a i n e d the periodic solution for the portions P0Q
a n d P4P~ of fig. 1 (i.e. without the QP7 part) and
followed the PaP~ b r a n c h until o~ = 0.06. H e also
I
20
'
I
40
'
I
'
BO
I
80
'
I
tOO
'
I
t2(~
'
I
t4
x
Fig. 5d. Flame front v(x) corresponding to fig. 5a.
f o u n d the soliton on fig. 4c. It turns out that this i:
n o t the only soliton-like solution. There are ii
effect two countable sets of values c ' ~ Csc
[0.835,0.86] and c ~ C~' c [0.48,0.50] for whic]
solitons exist! The curve in fig. 4a corresponds t,
c = inf Cs = 0.835, the one of fig. 4b to c = 0.84
a n d o n fig. 4c to c = sup Cs = 0.86. The respectiv
oblique flame fronts are shown on fig. 4e. O n fig.
the two solitons for c = inf C; -- 0.4845 and c :
sup C~' -- 0.49227 are displayed. Of course a refl~
tion with respect to the y = 0 axis produces anoth,
soliton which tends to the right critical point. L,
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
105
2--
2--
1-
O-
-1
-2
-3
-3
'
T
'
'
-I
-2
'
'
I
'
'
'
0
'
r
'
'
'
'
i
'
I
2
'
'
'
I
'
'
'
•
I
-t
-2
'
]
'
'
I
0
'
'
'
'
i
J
2
¥
¥
F i g . 6 a . O n e o s c i l a t i n g s o l u t i o n f o r c = 0.8.
Fig. 6b. An odd
asymptotic
solution (the half corresponding
to x_>0) for c=0.8.
0.5t"
0.0-
~-o.~-
0"
-i.O-
- t .5"
-2.0--~
-~ .5
,
-~.0
-0.5
0.0
0.5
i.0
Y
F i g . 7a. c ~ 0 . 5 9 ; T o r r o i d a l s o l u t i o n i n t h e elliptic r e g i o n P4Ps.
us explain in detail how these solitons are found.
Each soliton which leaves the left critical point,
leaves it by the one-dimensional unstable manifold. Locally this manifold is given by a converging power series. Using this series for the
difference scheme (4.l) we compute a triple
(Yj, Yj+I, Y j + 2 ) of points on the unstable manifold
and then continue with the scheme (4.1). Let
xm~,(c ) be the first point x where [y(x)[ >Ym~"
As with the odd solutions we are looking for local
maxima of the function Xm~(C). It turned out
again that these maxima are infinite, i.e. corre-
-2.0
,
,
,
i
-i.5
,
,
,
,
i
-~.0
,
,
,
,
i
-0.8
,
,
,
,
I
0.0
,
,
,
,
I
0.5
,
, ~ ]
t.O
¥.
F i g . 7 b . T h e s a m e a s i n fig. 7 a b u t in t h e
y'y"
plane.
spond to bounded solutions. Apparently, the respective set of values of c is a Cantor set K s, while
Cs consists of the boundary points of the "holes"
of K s i.e. Cs is the boundary of the complement of
K s. For c ~ K s \ C s the one dimensional unstable
manifold wanders in the space around the basic
patterns as on figs. 4a-c or fig. 8 and never reaches
back the critical point.
In the elliptic domains P4P5 and PrP7 the left
branch periodic solution is surrounded by invariant toil. Two such toil are shown on fig. 7a, b
and fig. 9a, b. As we pass through the point
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
106
i0l
0.5
0.0
y'
-0'5 l
-~..0
-t.5
-~.5
t , , , i , , , , L , , , , i , , , , i , , , , i L , , , I
-~..0
-0.5
0.0
0.5
~ .0
i.5
¥
Fig. 8. Two solitons: 1) for c = i n f C / - 0 . 4 8 4 5 ;
sup C" = 0.4982.
'1.0--
i.O--
0.5-
0,5-
0.0"
- 0.0.
>.-
-0.5-
2) for c =
-0.5-
-~..0
.00
-0.75
-0.50
-0.25
0.0
0.~5
0.50
0.75
~..00
-t .00
-0.75
-0.50
Y
-0.25
y,
0.00
0.25
0.50
0.7
Fig. 9a. c = 0.33; Torroidal solution in the elliptic region P6Pv.
Fig. 9b. The same as in fig. 9a.
P7 = P1 the maximal torus becomes more and more
thick so that for small c it looks as an egg with a
narrow hole running from one critical point to
another (see fig. 13a, b). One can easily prove that
the diameter of the set of bounded solutions tends
to zero as c ~ 0 so that at c = 0 the only bounded
solution is y-~ 0. An interesting phenomena is
displayed on figs. 12a-d. For the same c = 0.15
the solution on fig. 12c, d covers densely a torus
while the one on fig. 12a, b concentrates in a
spiral. It is known [16] that in a vicinity of a fixed
elliptic point a measure-preserving map P in a
generic situation has homoclinic points which cor-
respond to hyperbolic periodic points of the map
The center of the spiral on fig. 12a, b is apparentl,.
such a periodic solution while the spiral has seem
ingly a Cantor type cross-section caused by
presence of a homoclinic point.
At last we should mention the work of Rossle
[10]. In particular, Rossler computed some traje(
tories of the system
5c=-y-z,
p=x,
~=a(y-y2)-bz.
For b = 0 this system is equivalent to (1.5) wil
c = a / v ~ . On fig. 9. in [10] two trajectories of th
D. Michelson
/ Steady solutions of the Kuramoto-Sivashinsky equation
107
1.0--
0.5-
,f
,3
0.0-
-05
-10
!
25
50
75
~00
t25
150
~.75
X
Fig. 10a. c = 0.3; Three odd asymptotic solutions: 1) y(O); 2)
y(-2); 3) y ( 4 )
0.5--
0,5--
0.0-
0.0-
-0.5-
-0.5-
-1.0
'
-I.0
'
'
'
I
-0.5
'
'
'
'
I
'
'
'
'
0.0
I
'
'
'
0.5
'
-I.0
I
i.(
,
,
,
r
j
~
r
-0.5
-I .0
I
+0.0
Y
+
'
'
'
I
'
'
0,5
Y
Fig. 10b. Solution y(-2) in the yy' plane.
Fig, 10c+ Solution y(-4) in the yy' plane.
O. 5 0 - -
O. 25-
0.00-
-0.25-
-0.50-
-0.75
-0.8
'
I
-O.B
'
I
-0.4
'
I
-0.2
'
I
0.0
'
(
0.2
'
I
0.4
Y
Fig. 11. c = 0.2; Asymptotic solution y(0) and a periodic solution.
~
r
I
t .0
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
108
0.4--
0.6-
0.40.2-
0.20.0 -
~ .. 0 0 -0.2
I
~
-0.24
-4
-0.4-0.4-4
I
-0.6
'
-0
6
I
'
404
I
'
-0.2
I
'
0.0
I
'
02
t
'
0.4
I
-0.6
I
0.6
'
I
-0.4
-0.6
-0.2
Y
Fig. 12a. c = 0.15; A single o d d torroidal s o l u t i o n w i t h a s l o p e
y ' (0) = - 0.464.
0.4--
'
I
'
I
0.0
y'
'
I
0.2
Fig. 12b. T h e s a m e as in fig. 1 2 a b u t in the
0.4
y'y"
plane.
04--
0.20.2-
0.0.
O.r3>-
-0.2-
-0.2-0.4-
-o.6
'
I
-0.4
-06
'
i
-02
'
I
O0
'
I
02
'
I
04
~
"I
0.6
-04
1
-04
'
-06
'
I
-02
Y
Fig. 12c.
c=0.15;
analogous
to fig. 1 2 a b u t w i t h a s l o p e
r
y"
'
I
0.0
I
Fig. 12d. T h e s a m e as in 12c projected o n the
0
y'y"
y ' ( 0 ) = - 0.42.
04--
04--
0.2-
0.2-
~_o.o-
-0
- 0.0>-
-02-
2~
b
8
-0.4
-04
, , , , , i , , , , i , , , , 1 , ~ , , ! , , , ~ l , ~ , l t , , , i , ' , , , i
-0.3
-0.2
-0.:i
0.0
0.1
0.2
-04
Q.3
0.4
-0.3
-0.3
Y
Fig. 13a, b. c = 0.1; O n e torroidal solution.
-0.1
0.0
y,
O1
I
'
02
0,2
plane.
D. Michelson/ Steadysolutionsof the Kuramoto-Sioashinskyequation
system are displayed. In the first one c ~-0.1414
and the trajectory lies on a torus. In the second
one c = 0.2828 and the trajectory after several
spirals escapes to infinity. Rossler observes that
the invariant tori disappear at a---0.454 i.e. c
0.321. This indeed agrees with our results.
Appendix
Here we compute the Poincar6 map ~(/Jz, 8z'; e)
in (3.40) up to order O(e 2) and the Birkhoff normal form in (3.45).
First we solve the equation
z " + •(e)z' = e(1 - z : / 2 ) ,
5. Conclusion
Our analytical and numerical study has shown
that the set of steady solutions of the KuramotoSivashinsky equation is surprisingly complex.
There are conic solutions which correspond to the
Bunsen flames, oblique solitons and waves as well
as horizontal periodic, quasi-periodic and chaotic
solutions corresponding to a disturbed plane
flames. For a high propagation velocity only a
single conical solution exists, while for a lower one
all the above types of solutions do appear. Our
numerical study was devoted mainly to the odd
solutions. Certainly, more computations are in
place in order to understand the structure of the
set of all steady solutions and its dependence on
the propagation velocity c 2. However, a more important problem is the connection between the
time dependent solutions of (1.1) or (1.6) and the
above steady solutions. It was thought previously
that the turbulence in the Kuramoto-Sivashinsky
equation is primarily of a non-stationary origin
and is caused by a competition of a few spatial
nodes. In view of the above results it is plausible
that the set of steady solutions is an attractor for
the time dependent problem. Note that for the
(experimental) propagation velocity c 2 --1.2 of a
turbulent flame, both periodic orbits of (1.5) are
hyperbolic and there is plenty of spatial chaos in
the set of bounded solutions of (1.5). Thus the
turbulence in (1.1) may be attributed to the above
"steady" chaos. For large c numerical computations suggest that solutions of (1.6) (with proper
boundary conditions) tend to the unique conic
solution of (1.4). Since the conic solutions are
isolated also for small c, it would be interesting to
check whether they are locally stable.
109
)~(e) = 1 = O(e:)
(7.1)
in a neighborhood of the periodic solution
zper = - 2 s i n ~ -
E
g sin2~+ 0(e2).
Expand z as z = zpe~ + z o + ez 1 + 0(e2). Then
Zo" + z~ = 0,
Zo = al + a2 sin ~ + a 3 cos
(7.2)
z[" + z~ = 2 sin ~. z o - z2/2.
Thus
(1
/
a3
zl = - a l ~ s i n ~ + a2 ~+ g sin2~ + T cos2~
-
alaz~ sin ~ - ala3~ cos
+ (a2-a~)
12
c o s 2 ~ - a2a3 sin2~ 1
12
"
(7.3)
For the Poincar6 map we should impose
(7.4)
z"(o) = o
and
z"(~o) = 0,
where ~0 = 2~r + A~.
(7.5)
Then 9 a maps (z(0), z'(O)) into (Z(~o), z'(~o) ). By
(7.4).
a3=e(-2al
+ala2-
a---~)+O(e2).
(7.6)
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
110
while by (7.5) and (7.4)
brings ~ to the Birkhoff normal form
A~ = - z"(2rr
u 1 = Xu ei,,~(~)ua
)lz
= 0(e2)/(2-
"
(2~r)
a 2+ 0 ( ~ ) ) = O(e2).
(7.7)
+~20(lu13),
31r
~ ( ~ ) = - -6-~ + ~ ( ~ ) .
(7.12)
Thus
W i t h a cubic correction to u of order e one can
reduce the remainder in (7.12) to ezO([ul4 ).
Z(~o) - z(0) = z(Z~r) - z(0) + d~(e 2)
=
2~re(a2-a2/2-aZ/4)+(9(e
2)
and
References
z'(~0) - z'(o) = 2 ~ ( - a l + ~1~2/2) + ¢ ( , 2 ) .
[1] Y. Kuramoto and T. Tsuzuki, Persistent propagation of
concentration waves in dissipative media far from thermal
equilibrium, Prog. Theor. Phys. 55 (1976) 356-369.
[2] Y. Kuramoto and T. Yamada, Turbulent state in chemical
reaction, Prog. Theor. Phys. 56 (1976) 679.
[3] T. Yamada and Y. Kuramoto, A reduced model showing
chemical turbulence, Prog. Theor. Phys. 56 (1976) 681.
[4] Y. Kuramoto, Diffusion induced chaos in reaction systems, Suppl. Prog. Theor. Phys. 64 (1978) 346-367.
[5] G. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, I. Derivation of basic equations,
Acta Astronautica 4 (1977) 1117-1206.
[6] D. Michelson and G. Sivashinsky, Nonlinear analysis ot
hydrodynamic instability in laminar flames, II. Numerical
experiments, Acta Astronautica 4 (1977) 1207-1221.
[7] P. Manneville, Statistical properties of chaotic solutions oJ
a one-dimensional model for phase turbulence, Phys. Lett
84A (1981) 129-132.
[8] O.Yu. Tzverlodub, Stationary travelling waves on a filn
flowing down an inclined plane, Izvest. Akad. Nauk, Mekh
Zhid. Gaza (Russian), No. 4 (1980) pp. 142-146.
[9] T. Mitani, Stable solutions of non-linear flame shape equa
tion, Comb. Science and Technology 36 (1984) 235-247.
[10] O.E. Rossler, Continuous chaos-four prototype equ~
tions, Ann. N.Y. Acad. Sci. 316 (1979) 376-392.
[11] O. Thual and U. Frish, Natural boundary in Kuramot
model, presented at the workshop on Combustion, Flame
and Fires, March 1-15, 1984, Les Houches.
[12] B. Nicolaenko and B. Scheurer, Remarks on tl~
Kuramoto-Sivashinsky equation, Proceedings of the Col
ference on Fronts, Interfaces and Patterns, CNLS, L(
Alamos, May, 1983, Physica 12D (1984) 391-395,
[13] C. Siegel and J. Moser, Lectures on Celestial Mechanic
Grundlehren Bd. 187 (Springer, Berlin, 1971).
[14] J. Moser, On invariant curves of area-preserving mappin
of an annulus, Nach. Akad. Wiss. Gottingen, Math. Ph3
K1 (1962) 1-20.
[15] J. Moser, Stable and Random Motion in Dynamical S3
terns, with Special Emphasis on Celestial MechanJ
(Princeton Univ. Press, Princeton, 1973).
[16] E. Zehnder, Home,clinic points near elliptic fixed poin
Comm. Pure Appl. Math. 26 (1973) 131-182.
Since a I = 2(0) + O(e), a 2 = z'(0) - Zper(0) + O(e),
the map ~ is given by the formula
~ ( S z , ~z', e) -- (~z, 8z')
+2~re(3z'-
(6z)2/2-
(6z')Z/n;
- ~z + ~z- ~ ' / 2 ) + o ( ~ ) .
(7.s)
where 8z = z, 8z' = z' - z~e~(O). The differential
d ~ ( e ) has two conjugate eigenvalues
~ k = e ia°('),
]~--~-X,
ao(e)=Z~re+O(e2).
A linear transformation
w=3z'+i3z+O(e),
W=3z'-iSz+dg(e)
(7.9)
brings d ~ ( e )
comes
~(w,~,
to the diagonal form, while ~ be-
~) = ( w ~ , ~ ) ,
i~r,
w~ = Xw + --g- ( - w 2 + 3~ 2 - 6 w ~ ) + eZO(lwlZ).
(7.10)
T h e quadratic transformation
w=u+
~(-u2+6uFt-~2)+O(e)
(7.11)
D. Michelson/ Steady solutions of the Kuramoto-Sivashinsky equation
[17] C. Conley, Isolated Invariant Sets and the Morse Index,
Conf. Bd. Math. Sci., No. 38 (Amer. Math. Soc., Providence, 1978).
[18] N. Kopell and L.N. Howard, Bifurcations and trajectories
joining critical points, Advances in Math. 18 (1975)
306-358.
111
[19] M.S. Mock, On fourth-order dissipation and single conservation laws, Comm. Pure Appl. Math. 29 (1976)
383-388.
[20] C.K. McCord, Uniqueness of connecting orbits in the
equation y~3)=y2 _ 1, preprint (Dept. of Math., Univ. of
Wisconsin, Madison, 1983).